convex hull algorithms
TRANSCRIPT
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Formal Definitions(Convex Set)
• A set S is convex if x in S and y in S implies that the segment xy is a subset of S
• Example in 2D:
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Formal Definitions(Convex Hull of a Set of Points)
• The convex hull of a set S of points is the smallest convex set containing all the points in S
• Example in 2D:
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Intuitive Appreciations (Convex Hull of a Set of Points in 2D)
• The convex hull of a set of points in two dimensions is the shape taken by a rubber band stretched around nails pounded into the plane at each point
• Example:
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Intuitive Appreciations (Convex Hull of a Set of Points in 3D)
• The boundary of the convex hull of points in three dimensions is the shape taken by plastic wrap stretched tightly around the points
• Example:
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Applications • Collision Avoidance
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Applications • Minimum Bounding Box
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Applications • Shape Analysis
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Existing Convex Hull Algorithms • Gift Wrapping (1970)• Graham Scan (1972)• Quick Hull (1977)• Divide and Conquer (1977)• Monotone Chain (1979)• Incremental (1984)• Marriage before Conquest (1986)• Chan (1996)
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Interior Points Algorithm
Based on the following Lemma
A point is non-extreme if and only if it is inside some (closed) triangle whose vertices are points of the set and is not itself a corner of that triangle
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Interior Points Algorithm
Algorithm: INTERIOR POINTS
for each i do
for each j != i do
for each k != i != j do
for each l != k != i != j do
if p(l) in Triangle{ p(i), p(j), p(k) }
then p(l) is non-extreme
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Any Questions?
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Thank You!